Lecture 9Symmetric Matrices

Subspaces and Nullspaces

Shang-Hua Teng

Matrix Transpose

• Addition: A+B

• Multiplication: AB

• Inverse: A-1

• Transpose : AT

jiijT AA

Transpose

84

73

62

51

8765

4321T

5

4

3

2

1

54321 T

Inner Product and Outer Product

703221125

8

7

6

5

4321

3224168

2821147

2418126

2015105

4321

8

7

6

5

Properties of Transpose

11

TT

TTT

TTT

AA

ABAB

BABA

End of Page 109: for a transparent proof

Ellipses and Ellipsoids

122

r

y

R

x

1/10

0/1,

2

2

y

x

r

Ryx

R

r

0

1

/10

00

0/1 1

2

21

1

nn

n

x

x

r

r

xx

Later

R

r 0

yAxyAx TTT

Relating to

Symmetric Matrix

• Symmetric Matrix: A= AT

1;John2:Alice

4:Anu3:Feng

Graph of who is friend with whomand its matrix

1101

1110

0111

1011

Examples of Symmetric Matrices

n

TTT

d

d

D

BDBBBBB

1

, ,

B is an m by n matrix

Elimination on Symmetric Matrices

• If A = AT can be factored into LDU with no row exchange, then U = LT. In other words

The symmetric factorization of a symmetric matrix is A = LDLT

10

21

40

01

12

01

82

21

So we know Everything about Solving a Linear System

• Not quite but Almost

• Need to deal with degeneracy (e.g., when A is singular)

• Let us examine a bigger issues:

Vector Spaces and Subspaces

What Vector Spaces Do We Know So Far

• Rn: the space consists of all column (row) vectors with n components

nRRRR ,,,, 321

Properties of Vector Spaces

xxxxxx

cycxyxcbyaxxba

bxaxabzyxzyx

xyyx

00)(

0x0 ;1 ;)(

)( );()(

)()( ;

Other Vector Spaces

matrices by real all ofset the:M

: functions real all ofset the:F

0 :Z

nm

RRxf

Vector Spaces Defined by a Matrix

nRxAxAC :)(

For any m by n matrix A

• Column Space:

• Null Space: 0:)( AxxAN

222

111

010

01

N

N

General Linear System

The system Ax =b is solvable if and only if b is in C(A)

Subspaces

• A subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: if v and w are vectors in the subspace and c is any scalar, then– v+w is in the subspace– cv is in the subspace

Subspace of R3

• (Z): {(0,0,0)}

• (L): any line through (0,0,0)

• (P): any plane through (0,0,0)

• (R3) the whole space

A subspace containing v and w must contain all linear combination cv+dw.

Subspace of Rn

• (Z): {(0,0,…,0)}• (L): any line through (0,0,…,0)• (P): any plane through (0,0,…,0)• …• (k-subspace): linear combination of any k independent

vectors • (Rn) the whole space

Subspace of 2 by 2 matrices

a

a

d

a

d

ba

0

0 :I of multiple all ofset the

0

0 matrices diagonal :D

0 matricesngular upper tria :U

00

00 :Z

Express Null Space by Linear Combination

• A = [1 1 –2]: x + y -2z = 0

x = -y +2z

Free variablesPivot variable

• Set free variables to typical values

(1,0),(0,1)

• Solve for pivot variable: (-1,1,0),(2,0,1)

{a(-1,1,0)+b(2,0,1)}

Express Null Space by Linear Combination

2032

3121A

Guassian Elimination for finding the linear combination: find an elimination matrix E such that

EA = free

pivot

Permute Rows and Continuing Elimination (permute columns)

011121

131111

021111

110011

A

Theorem

If Ax = 0 has more unknown than equations (m > n: more columns than rows), then it has nonzero solutions.

There must be free variables.

Echelon Matrices

*

*

*

*

000000

**0000

*****0

******

A

Free variables